A194801 - OEIS

%I #55 Aug 24 2024 09:45:32

%S 0,0,1,0,1,0,0,1,1,3,0,1,2,4,1,0,1,3,5,4,6,0,1,4,6,7,9,3,0,1,5,7,10,

%T 12,9,10,0,1,6,8,13,15,15,16,6,0,1,7,9,16,18,21,22,16,15,0,1,8,10,19,

%U 21,27,28,26,25,10,0,1,9,11,22,24,33,34,36,35

%N Square array read by antidiagonals: T(n,k) = k*((n+1)*k-n+1)/2, k = 0, +- 1, +- 2,..., n >= 0.

%C Note that a single formula gives several types of numbers. Row 0 lists 0 together the Molien series for 3-dimensional group [2,k]+ = 22k. Row 1 lists, except first zero, the squares repeated. If n >= 2, row n lists the generalized (n+3)-gonal numbers, for example: row 2 lists the generalized pentagonal numbers A001318. See some other examples in the cross-references section.

%e Array begins:

%e (A008795): 0, 1,  0,  3,  1,  6,  3, 10,   6,  15,  10...

%e (A008794): 0, 1,  1,  4,  4,  9,  9, 16,  16,  25,  25...

%e A001318:   0, 1,  2,  5,  7, 12, 15, 22,  26,  35,  40...

%e A000217:   0, 1,  3,  6, 10, 15, 21, 28,  36,  45,  55...

%e A085787:   0, 1,  4,  7, 13, 18, 27, 34,  46,  55,  70...

%e A001082:   0, 1,  5,  8, 16, 21, 33, 40,  56,  65,  85...

%e A118277:   0, 1,  6,  9, 19, 24, 39, 46,  66,  75, 100...

%e A074377:   0, 1,  7, 10, 22, 27, 45, 52,  76,  85, 115...

%e A195160:   0, 1,  8, 11, 25, 30, 51, 58,  86,  95, 130...

%e A195162:   0, 1,  9, 12, 28, 33, 57, 64,  96, 105, 145...

%e A195313:   0, 1, 10, 13, 31, 36, 63, 70, 106, 115, 160...

%e A195818:   0, 1, 11, 14, 34, 39, 69, 76, 116, 125, 175...

%o (GW-BASIC)

%o 100 'SQUARE ARRAY T(N,K) A194801

%o 200 DIM T(11,10)

%o 210 FOR N=0 TO 11

%o 220  FOR J=0 TO 5

%o 230    IF J>0 THEN T(N,K)=  J *((N+1)*  J -N+1)/2: K=K+1

%o 240                T(N,K)=(-J)*((N+1)*(-J)-N+1)/2: K=K+1

%o 250  NEXT J

%o 260  K=0

%o 270 NEXT N

%o 299 '...

%o 300 'PRINT SQUARE ARRAY T(N,K) (see example)

%o 310 FOR N=0 TO 11

%o 320  FOR K=0 TO 10

%o 330   PRINT T(N,K);

%o 340  NEXT K

%o 350  PRINT

%o 360 NEXT N

%o 500 END

%Y Rows (0-11): 0 together with A008795, (truncated A008794), A001318, A000217, A085787, A001082, A118277, A074377, A195160, A195162, A195313, A195818

%Y Columns (0-9): A000004, A000012, A001477, (truncated A000027), A016777, (truncated A008585), A016945, (truncated A016957), A017341, (truncated A017329).

%Y Cf. A139600.

%K nonn,tabl

%O 0,10

%A _Omar E. Pol_, Feb 05 2012