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By N. J. A. Sloane

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Drawn next to a mirror. (See Fig. ) (15) Ο,Τ,Τ, F, F,S,S,E,N,T,E,T,T,F,F,S,S,E,N,T,T,T,T the initial letters of the English names for the numbers. M Ώ 8 H δ... Fig. 26. 16 A puzzle. SEQUENCES FROM LATTICE STUDIES IN PHYSICS In the last twenty years physicists have studied a number of basic combinatorial problems related to crystal lattices. Typical problems are to find the number of self-avoiding paths of length n on a given lattice, or the number of ways a particular graph can be drawn on the lattice.

And [β], [2β], [3/3],... together contain all the positive integers without repetition, where [x] denotes the greatest integer less than or equal to x. 61803. . gives seqs. 917: 1, 3, 4, 6, 8, 9, . . and 509: 2, 5, 7, 10, 13, 15, The following test for Beatty sequences is due to R. L. Graham. If ax ,a2,. . is a Beatty sequence, then the values of al,. . , αη-λ determine 29 /// SOME IMPORTANT SEQUENCES an to within 1. Look at the sums ax + an-x, a2 + an-2, . . , an-x + ax. If all these sums have the same value, V say, then an must equal V or V + 1 ; but if they take on the two values V and V + 1, and no others, then an must equal V + 1.

211 1,2,3,4,6, 16, 16,30 POINT-SYMMETRIC TOURNAMENTS. REF CMB 13 322 70. 212 1,2,3, 4, 7, 13, 24, 44, 83, 157, 297, 567, 1085, 2086, 4019, 7766, 15039, 29181, 56717,110408,215225, 420076, 820836, 1605587, 3143562, 6160098, 12080946 LANDAUS APPROXIMATION. REF MTAC 18 79 64. 213 1,2,3,4,8, 10, 12, 19,37,54,65,77,314,366,770, 1327, 1373, 1937,2561, 2663,5834, 5985,6751, 12003 SIZE OF EVEN PERFECT NUMBERS. REF BE3 19. NAMS 18 608 71. 214 1, 2, 3, 4, 9, 27, 512, 134217728, (NEXT TERM HAS 155 DIGITS) AN EXPONENTIAL FUNCTION ON PARTITIONS.

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